Departments of Mathematics
Montana State University
Fall 2015
Prof. Kevin Wildrick
An introduction to non-smooth analysis and geometry
Lecture 2: Metric measure spaces
In this lecture, we will discuss the basic structure of the spaces we are interested in
studying. The key structures are drawn from the structure of Riemannian manifolds that
have a chance of being preserved under limits: metric and measure.
1. The definitions and some examples
1.1. Measures. In modern analysis, the role of measurability is much smaller than in a
classical measure theory course. For this reason, we will adopt the following somewhat
non-standard definition of the word “measure”.
Definition 1.1. Let X be a topological space. A measure on X is a function µ : P(X) →
[0, ∞] such that
• µ(∅) = 0,
• if {Ei }i∈N is a countable collection of subset of X, then
!
[
X
µ
Ei ≤
µ(Ei ),
i
i∈N
• if U ⊆ X is an open set and E ⊆ X is any set, then
µ(E) = µ(E ∩ U ) + µ(E\U ),
i.e., open sets are measurable,
• for any set E ⊆ X, there is a Borel set U ⊇ E such that
µ(E) = µ(U ),
i.e., µ is Borel regular.
• each point x ∈ X has a neighborhood U with µ(X) < ∞, i.e., µ is locally finite,
• there is a set U ⊆ X such that µ(X) 6= 0, i.e., µ is non-trivial.
In this course, we will not concern ourselves with measurability issues, as they are
tedious and can in practice almost always be overcome with some standard techniques.
1.2. Metric spaces.
Definition 1.2. A metric space is a pair (X, d) where X is a set and d : X × X → [0, ∞)
is a function such that for all x, y, z ∈ X,
• d(x, y) = 0 if and only if x = y (non-degeneracy),
• d(x, y) = d(y, x) (symmetry),
• d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
For x ∈ X and r > 0, we define the open ball at x of radius r by
B(x, r) := {y ∈ X : d(x, y) < r},
and for r ≥ 0, the closed ball at x of radius r by
B(x, r) := {y ∈ X : d(x, y) ≤ r}.
The diameter of a set E ⊆ X is defined
diam(E) := sup{d(x, y) : x, y ∈ E}.
1
We will use the convention that a metric space (X, d) is always equipped with the topology
induced by the metric, namely, the smallest topology in which the open balls are open.
Be careful, however, with the terminology: the topological closure of an open ball B(x, r)
might be different from B(x, r).
Metric spaces are extremely common in mathematics. We give here only a few examples
- we’ll see many more during the semester.
• Let V be a vector space equipped with an inner product h·, ·i. Then
||v|| := (hv, vi)1/2
is a norm, and d(v, w) = ||v −w|| is a metric. The standard metric on Rn is defined
in this way.
• Let V be a vector space equipped with a norm. Then d defined as before still
defines a metric. For 1 ≤ p ≤ ∞, the p-norm on Rn defines a metric in this way.
• Let M be a smooth manifold equipped with a smoothly varying inner product g
on its tangent spaces. Let I be a compact interval in R. For a continuous mapping
γ : I → M (a path), define the g-length of γ by
Z
length(γ) = ||γ 0 (s)||2g ds.
g
I
For points p and q of M , define
dg (p, q) = inf length(γ),
g
where the infinimum is taken over all paths starting at p and ending at q. Then dg
is a metric on M . Unfortunately, both g and dg are referred to as a Riemannian
metric on M .
• In the previous example, we started with a notion of length and defined a metric.
This can be reversed as well. Let (X, d) be any metric space, and let γ : I → X
be a path in X. Then
n
X
length(γ) := sup
d(γ(si−1 ), γ(si ))
d
i=1
where the infimum is taken over all partitions s0 , . . . , sn of I. If (X, d) has the
property that any two points can be connected by a path of finite length, then
e y) = inf length(γ)
d(x,
d
is a metric on X, where the infimum is taken over all paths connecting x to y.
• If (X, d) is a metric space and S ⊆ X is a subset, then d|S×S defines a metric on
S.
• If (X1 , d1 ) and (X2 , d2 ) are metric spaces, and || · ||p denote the p-norm on R2 , then
d((x1 , x2 ), (y1 , y2 ) = ||(d1 (x1 , y1 ), d2 (x2 , y2 ))||p
defines a metric on X1 × X2 .
• Let (X, d) be any metric space, and let 0 < α < 1. Then
dα (x, y) := (d(x, y))α
defines a metric on X called the α-snowflaked metric. If (X, d) Note that dα is
only sometimes a metric if α > 1.
• Let (X, d) be a metric space, let Y be a topological space, and let f : X → Y be
a homeomorphism. Then
dY (a, b) := d(f −1 (a), f −1 (b))
defines a metric on Y .
1.3. Metric Measure Spaces.
Definition 1.3. A metric measure space is a triple (X, d, µ) where (X, d) is a separable
metric space, and µ is a measure on X.
1.4. Examples of metric measure spaces. Examples of metric measure spaces are
found in nearly all parts of mathematics. Here we give just a few that indicate the
historical motivations for this course come.
• Euclidean space Rn with its standard metric and Lebesgue measure,
• Any finite dimensional normed vector space with the associated metric and Lebesgue
measure,
• Any orientend Riemannian manifold with its geodesic metric and the measure
associated to its volume form,
• Sub-Riemannian manifolds and Carnot groups (definitions later)
• Finite dimensional simplicial complexes equipped with a path metric and the
highest-dimensional Lebesgue measure
• plenty of “ad hoc” constructions
Perhaps the most general and important construction of a metric measure space comes
from geometric measure theory.
Definition 1.4. Let (X, d) be a metric space, and let 0 ≤ s < ∞, and let δ > 0. The
s-dimensional Hausdorff δ-content, of a set E ⊆ X is defined by
(
)
X
[
Hδs (E) = inf
(diam Ek )s : Ek ⊆ X, diam Ek < δ, E ⊆
Ek .
k∈N
k∈N
Hδs (E)
Then
is a non-increasing function of δ (possibly a constant function with value ∞
or 0). Hence
Hs (E) := lim Hδs (E)
δ→0
is well defined; this is called the s-dimensional Hausdorff measure of E.
If the s-dimensional Hausdorff measure of (X, d) is finite and positive, then (X, d, Hs )
is a metric measure space.
An important class of examples that arise in this way (but not in any of the other ways
mentioned above) consists of boundaries of hyperbolic groups.